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Chapter 8: Problem 43

Evaluate the given expression for \(x=-3\) and \(y=5\) $$x^{-2}$$

Short Answer

Expert verified
The value of \(x^{-2}\) for \(x = -3\) is \(\frac{1}{9}\).

Step by step solution

01

- Understand the Expression

The expression to evaluate is \(x^{-2}\). This indicates that we need to find the value of \(x^{-2}\) when \(x = -3\).
02

- Substitute the Value of x

Substitute \(x = -3\) into the expression: \( (-3)^{-2} \).
03

- Apply Negative Exponent Rule

Using the negative exponent rule, \(a^{-n} = \frac{1}{a^n}\), rewrite the expression: \((-3)^{-2} = \frac{1}{(-3)^2} \).
04

- Calculate the Positive Exponent

Calculate \((-3)^2 \): \((-3)^2 = 9 \).
05

- Evaluate the Fraction

Substitute \9\ into the fraction: \(\frac{1}{9} \). Thus, \((-3)^{-2} = \frac{1}{9} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

negative exponents
Negative exponents can often seem tricky at first. But understanding how they work is essential for evaluating and simplifying expressions. A negative exponent means you need to take the reciprocal of the base raised to the corresponding positive exponent. For example, the expression \( x^{-2} \) is the same as \( \frac{1}{x^2} \). Essentially, the negative exponent tells you to flip the base to the denominator and then apply the positive exponent. Think of it as 'flipping' the base to the bottom of a fraction.
substitution
In mathematics, substitution involves replacing variables with their given numerical values to simplify or evaluate expressions. For this exercise, you substitute \( x = -3 \) into the expression \( x^{-2} \). By doing so, \( x^{-2} \) becomes \( (-3)^{-2} \), allowing you to work with a specific number rather than a variable.
exponent rules
Exponent rules are crucial for simplifying expressions and solving equations. One of the key rules is the negative exponent rule: \( a^{-n} = \frac{1}{a^n} \). This rule shows that you can convert a negative exponent into a positive exponent by taking the reciprocal of the base. Additionally, understanding how to multiply and divide powers with the same base can help you simplify expressions further. For example, \( a^m \times a^n = a^{m+n} \) and \( a^m \div a^n = a^{m-n} \). These rules make it easier to manage and evaluate complex expressions.
simplifying expressions
Simplifying expressions is about combining like terms and reducing the expression to its simplest form. When dealing with exponents, it often involves applying exponent rules and basic arithmetic. In the given problem, \( (-3)^{-2} \) is simplified first by applying the negative exponent rule: \( (-3)^{-2} = \frac{1}{(-3)^2} \). Then, you further simplify it by calculating the positive exponent: \( (-3)^2 = 9 \). Finally, you substitute this value into the fraction to get \frac{1}{9}, simplifying the initial expression completely.

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Most popular questions from this chapter

Perform the indicated operations and simplify. $$\left(5 x^{2} y\right)^{2}$$

Factor each of the following as completely as possible. If the polynomial is not factorable, say so. $$-w^{2}-16$$

Divide each of the following. Use the long division process where necessary. $$\frac{20 t^{3}+33 t^{2}-4}{5 t+2}$$

Factor each of the following as completely as possible. If the expression is not factorable, say so. Try factoring by grouping where it might help. $$6 u^{3} v^{2}+18 u^{3} v^{3}-12 u^{3} v^{5}$$

Divide each of the following. Use the long division process where necessary. $$\frac{2 x^{2}-3 x+7}{x-1}$$
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